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Advanced techniques

Nonelementary primitives and special functions

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

∫e−x2 dx=π2erf⁡(x)+C\int e^{-x^2}\,dx=\frac{\sqrt\pi}{2}\operatorname{erf}(x)+C

Some integrals have no elementary primitive. They can be represented using erf, exponential integrals or elliptic integrals. Definite values can also use approximations with error control.

Common mistake

Do not confuse absence of an elementary primitive with absence of an integral.

Step-by-step example

  1. Define the error function.

    erf⁡(x)=2π∫0xe−t2 dt\operatorname{erf}(x)=\frac2{\sqrt\pi}\int_0^xe^{-t^2}\,dt

  2. Differentiate using the fundamental theorem.

    erf⁡′(x)=2πe−x2\operatorname{erf}'(x)=\frac2{\sqrt\pi}e^{-x^2}

  3. Adjust the factor and add C.

    I=π2erf⁡(x)+CI=\frac{\sqrt\pi}{2}\operatorname{erf}(x)+C

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